Squeeze Theorem: Simplify Limits of Functions with Sandwich Approach – A Guide to Using the Squeeze Theorem in Calculus

Squeeze or Sandwich Theorem

The squeeze (or sandwich) theorem states that if f(x)≤g(x)≤h(x) for all numbers, and at some point x=k we have f(k)=h(k), then g(k) must also be equal to them

The squeeze or sandwich theorem, also known as the squeeze or pinching theorem, is a result in mathematics that is used to find the limit of a function. It provides a method to evaluate limits by exploiting the squeeze property of functions.

The squeeze theorem states that if f(x) ≤ g(x) ≤ h(x) for all x in a neighborhood of a, and if lim f(x) = lim h(x) = L, then lim g(x) = L as well. In other words, if there are two functions that squeeze a third function between them, and the limits of the two squeezing functions are the same, then the limit of the squeezed function must be the same as well.

This theorem is particularly useful in situations where it is difficult to evaluate the limit of a function directly. By finding two other functions that bound the function in question, and whose limits can easily be found, we can use the squeeze theorem to find the limit we are interested in. This theorem is commonly used in calculus, especially in the study of limits of sequences and series.

For example, consider the limit of sin(x)/x as x approaches 0. This limit is not immediately obvious, and it is difficult to evaluate directly. However, we can use the squeeze theorem to find it by finding two other functions that bound sin(x)/x from above and below. We know that sin(x) ≤ x for all x, so sin(x)/x ≤ 1 for all x ≠ 0. Additionally, we know that lim x → 0 sin(x) = 0 and lim x → 0 1 = 1, so by the squeeze theorem, lim x → 0 sin(x)/x = 0.

Overall, the squeeze or sandwich theorem is a powerful tool in the study of limits of functions. It allows us to evaluate limits in situations where direct evaluation is not possible, and it can make complex limits much simpler to handle.

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